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Stochastic Lifestyling: Optimal Dynamic Asset Allocation
for Defined Contribution Pension Plans
Andrew J.G. Cairnsab, David Blakec, Kevin Dowdd
First version: June 2000This version: September 13, 2004
Abstract
This paper considers the asset-allocation strategies open to members of defined-contribution pension plans. We investigate a model that incorporates three sourcesof risk: asset risk and salary (or labour-income) risk in the accumulation phase;and interest-rate risk at the point of retirement. We propose a new form of termi-nal utility function, incorporating habit formation, that uses the plan member’sfinal salary as a numeraire. The paper discusses various properties and character-istics of the optimal stochastic asset-allocation strategy (which we call stochasticlifestyling) both with and without the presence of non-hedgeable salary risk. Wecompare the performance of stochastic lifestlying with some popular strategiesused by pension providers, including deterministic lifestyling (which involves agradual switch from equities to bonds according to preset rules) and static strate-gies that invest in benchmark mixed funds. We find that the use of stochasticlifestyling significantly enhances the welfare of a wide range of potential planmembers relative to these other strategies.
JEL classification: E21, G11, G23.
Keywords: stochastic control; optimal asset allocation; stochastic lifestyling; util-ity numeraire; habit formation; non-hedgeable salary risk; HJB equation.
aActuarial Mathematics and Statistics, School of Mathematical and Computer Sciences,Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom
bCorresponding author: E-mail [email protected] Business School, City University, 106 Bunhill Row, London, EC1Y 8TZ, United King-
domdCentre for Risk & Insurance Studies, Nottingham University Business School, Jubilee Cam-
pus, Nottingham, NG8 1BB, United Kingdom
1
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1 Introduction
A popular asset allocation strategy for managing equity risk during the accumula-tion phase of a defined contribution (DC) pension plan is deterministic lifestyling.At the beginning of the plan, the contributions are invested entirely in equities.Then, beginning on a predetermined date (e.g., ten years) prior to retirement,the assets are switched gradually into bonds at a rate equal to the inverse of thelength of the switchover period (e.g., 10% per year). By the date of retirement, allthe assets are held in bonds, which are then sold to purchase a life annuity thatprovides the pension. The aims of the strategy are to reduce the impact on thepension of a catastrophic fall in the stock market just before the plan member re-tires and to hedge the interest-rate risk inherent in the annuity-purchase decision.Deterministic lifestyling is a simple strategy to explain to plan members and toimplement, and is widely used as the default strategy or as one option offered bymany UK DC pensions providers. Similar deterministic strategies have also beenrecommended in other countries (for example, in a US context, Malkiel, 2003,recommends a mix of bonds and equities which changes over time in a similarway to the deterministic lifestyle strategy). However, there is no evidence that itis an optimal strategy in an objective sense.
The purpose of this paper is to find the optimal dynamic asset allocation strategyfor a defined contribution pension plan, taking into account the stochastic featuresof the plan member’s lifetime salary progression as well as the stochastic propertiesof the assets held in his accumulating pension fund. Of particular importanceis the fact that salary risk (or labour-income risk: the fluctuation in the planmember’s earnings in response to economic shocks) is not fully hedgeable usingexisting financial assets. To illustrate, wage-indexed bonds could be used to hedgeboth productivity and inflation shocks, but such bonds are not widely traded.The paper builds on Blake, Cairns & Dowd (2001) which developed a pensionplan accumulation programme designed to deliver a retirement pension that isclosely related to the salary (and hence standard of living) that the plan memberreceived immediately prior to retirement. We call the optimal dynamic assetallocation strategy stochastic lifestyling and compare it against various static anddeterministic lifestyle strategies to calculate the cost of suboptimal strategies.Moreover, stochastic lifestyling is still a relatively easy strategy to implement inpractice, despite the apparent increase in complexity compared to deterministiclifestyling.
The solution technique uses the expected present value of future contributionpremiums into the plan. This is not a new idea and has been used by Boulier etal. (2001), Deelstra et al. (2000) and Korn & Krekel (2002) and others, building onthe original work of Merton (1969, 1971). Liu (2001) examines ways in which theMerton framework can be generalised to include, for example, stochastic interestrates and stochastic risk premia, but only for the case where utility is a function
2
of the cash lump sum at the beginning of retirement.
Where our approach differs from these studies is in:
• the use of a salary-related numeraire (or utility numeraire) as an argumentin the plan member’s utility function1; and
• assuming that the purpose of the pension plan is to deliver a pension (i.e.life annuity) in retirement rather than a cash lump sum at the date ofretirement2.
Although these differences do not alter the basic form of the optimal solutionderived in these earlier studies, we find that the optimal proportions invested ineach of the key asset classes, cash, bonds and equities, are very different. Moresignificantly, we also find that these optimal proportions often differ substantiallyfrom those implied by deterministic lifestyling (which ignores both the plan mem-ber’s attitude to risk and any correlation between his salary and the returns onassets held in the fund), so that the cost of the latter strategy can be considerablein terms of the additional premiums into the plan needed to match the expectedutility of the optimal strategy.
However, unlike the case for deterministic lifestyling, the optimal asset allocationunder stochastic lifestyling is sensitive to certain underlying assumptions, e.g.,concerning the process determining interest rates. To clarify key issues, we there-fore first derive our results using a simple stochastic model in which the interestrate is deterministic (Section 2). We then extend the model to a more generalstochastic setting (Section 3). This allows us to analyse separately (a) the effect ofthe salary-related numeraire in the utility function and (b) the pension purchasedat retirement and its dependence on uncertain interest rates.
We show that in the former case the optimal asset allocation can be replicatedusing two efficient mutual funds, whereas the latter case needs three efficientmutual funds. One mutual fund (which is heavily dominated with equities) isdesigned to satisfy the risk appetite of the plan member. The second fund (whichis heavily dominated with cash) is designed to hedge the salary risk within thepension plan. The third fund (which is heavily dominated with bonds) is designedto hedge interest rate (and hence annuity) risk in the case where interest rates arestochastic.3
1In the studies of Boulier et al., 2001, and Deelstra et al., 2000, utility is also measured relativeto a salary-related benchmark, but the utility function depends on the monetary surplus overthis benchmark, rather than on the surplus as a proportion of final salary as here.
2In countries, such as the UK, it is mandatory to use the cash lump sum to purchase a lifeannuity by a certain age, while in other countries, such as the US, it is not.
3Parts of the problem are related to previous analyses of defined-benefit (DB) pension plans(Sundaresan and Zapatero, 1997, and Cairns, 2000). Both the DC and DB problems require the
3
2 A simple stochastic model
2.1 The structure of the model
This section develops the optimal asset allocation strategy using a simple modelwith deterministic nominal interest rates. The aim of the simple model is tohighlight the following features:
• the use of the plan member’s salary as a numeraire in the utility function;
• the treatment of a stream of contribution premiums linked to salary; and
• the consequences of salary not being fully hedgeable.
The first two features are straightforward to deal with and do not cause particularproblems when salary is fully hedgeable (that is, when there is a complete market).The third feature, however, implies that the market is incomplete and so givesrise to qualitatively different results from the complete market case.
The structure of the simple model is as follows:
• There are two underlying assets in which the pension plan can invest: onerisk free (a cash fund) and one risky (an equity fund). The risk-free asset hasa price R0(t) = R0(0) exp(rt) at time t, where r is the constant risk-free nom-inal rate of interest. The risky asset has price R1(t) at time t and satisfies thestochastic differential equation (SDE) dR1(t) = R1(t) [(r + ξ1σ1)dt + σ1dZ1(t)] ,where Z1(t) is a standard Brownian motion and ξ1 and σ1 are constants. Therisk premium on this asset is ξ1σ1, where ξ1 is the market price of risk. Thesolution for R1(t) is
R1(t) = R1(0) exp
[(r + ξ1σ1 − 1
2σ2
1
)t + σ1Z1(t)
]. (2.1.1)
• The pension plan member has a salary at time t of Y (t). Y (t) is governedby the SDE dY (t) = Y (t) [(r + µY )dt + σY 0dZ0(t) + σY 1dZ1(t)] , where µY
is a constant and Z0(t) is a second standard Brownian motion independentof Z1(t). The σY 1 term allows for possible correlation between the salary
determination of an optimal asset-allocation strategy using stochastic control methods. However,typical DB problems involve optimising the sponsor’s utility, whereas here we are optimisingthe individual plan member’s utility. The DB model of Sundaresan and Zapatero, 1997, sharessome characteristics with the DC models of Deelstra et al. (2000) and Boulier et al. (2001).These three papers incorporate a DB type of guarantee, and all three analyses rely upon salaryrisk being fully hedgeable. However, in the DB plan surplus over the guarantee reverts to theplan sponsor while in the DC plans the full value of the assets goes to the member.
4
and equity returns (for example, salary might be related to the profitabilityof the company or the general state of the economy). Y (t) has the solution
Y (t) = YH(t)YN(t) (2.1.2)
where YH(t) = Y (0) exp
[(r + µY − 1
2σ2
Y 1
)t + σY 1Z1(t)
]
and YN(t) = exp
[−1
2σ2
Y 0 + σY 0Z0(t)
].
YH(t) is the hedgeable component of Y (t) and YN(t) is the non-hedgeablecomponent.
• Premiums are payable in one of two forms: either (a) as regular premiums,i.e., continuously into the plan member’s individual account at the rate ofπY (t), so that premiums are a constant proportion, π, of salary4, or (b) asan initial single premium, in which case π = 0.
• The value of the plan member’s account (i.e., his pension wealth) is denotedby W (t). If a (possibly time-dependent) proportion p(t) of this account isinvested in the risky asset then the dynamics of W (t) are governed by theSDE
dW (t) = W (t) [(r + p(t)ξ1σ1)dt + p(t)σ1dZ1(t)] + πY (t)dt.
• The plan member will retire at time T , is risk averse, and has a terminalutility function, u(W (T ), Y (T )), that depends both on terminal pensionwealth and terminal income
u(W (T ), Y (T )) =
1γ
(W (T )Y (T )
)γ
where γ < 1 and γ 6= 0
log(
W (T )Y (T )
)when γ = 0.5
• The plan member’s objective is to find the optimal dynamic asset alloca-tion strategy, namely the weight in equities, p(t), to maximise his expectedterminal utility.
4The contribution rate π is given exogeneously, as is typical in many occupational DC plans.Also implicit in this model is the assumption that the rate of consumption before retirement isgiven exogenously and is equal to (1−π)Y (t); in other words, there are no non-pensions savingsin the model.
5In the case of log utility, E[log W (T )/Y (T )] = E[log W (T )]−E[log Y (T )]. Since E[log Y (T )]is fixed, the optimal investment strategy will be the same for this case as for problems wherethe utility function is just log W (T ) (i.e., the utility function involves just a cash numeraire).
5
The new numeraire is straightforward to justify. At the time of retirement, theplan member will be concerned about the preservation of his standard of living. Sohe will be interested in his retirement income relative to his pre-retirement salary.This is consistent with consumption-smoothing features of the life-cycle modelof Ando and Modigliani (1963). It is also consistent with the habit-formationmodel developed by Ryder and Heal (1973), Sundaresan (1989) and Constan-tinides (1990). Specifically, the plan member’s utility at retirement is directlyrelated to his or her recent salary, Y (T ), or consumption level, given exogenouslyas (1− π)Y (T ).
Defining a new state variable X(t) ≡ W (t)/Y (t) allows terminal utility to bewritten u(X(T )) = γ−1X(T )γ. A straightforward application of Ito’s formulashows that the SDE for X(t) is
dX(t) =[π + X(t)
(−µY + p(t)σ1(ξ1 − σY 1) + σ2Y 0 + σ2
Y 1
)]dt
−σY 0X(t)dZ0(t) + X(t)(p(t)σ1 − σY 1)dZ1(t). (2.1.3)
From equation (2.1.3) we can see that X(t) provides us with all the informationwe require to solve the problem. Additional information on W (t) and Y (t) willnot alter the ultimate distribution for X(T ). In the remainder of this section,therefore we will focus on X(t) alone.
The optimisation problem can now be stated as: Maximise, over all admissibleasset allocation stategies, the expected terminal utility (that is: maximise, overp(t), E[u(X(T ))]).
We will now consider four separate cases:
• Case 1: π = 0, σY 0 = 0.
• Case 2: π = 0, σY 0 6= 0.
• Case 3: π > 0, σY 0 = 0.
• Case 4: π > 0, σY 0 6= 0.
The first two cases indicate zero contributions to the pension fund, and wouldapply to a plan member who has accumulated some pension wealth and has chosennot to save any more towards retirement. The latter two cases indicate positiveongoing contributions. The first and the third cases indicate zero non-hedgeablesalary risk and the second and fourth cases indicate the existence of non-hedgeablesalary risk.
6
2.2 Case 1: π = 0, σY 0 = 0, and Case 2: π = 0, σY 0 6= 0
We will first consider Case 2. Case 1 can then be solved by taking σY 0 = 0. InCase 2, the salary numeraire cannot be replicated with existing financial assets,since the market is incomplete. Nevertheless we can still derive an analyticalsolution to the problem. The optimal expected utility (the value function) is
V (t,X(t)) = h(t)X(t)γ,
where
h(t) =1
γexp [γθ2(T − t)]
and θ2 =γ + 1
2σ2
Y 0 +(ξ1 − σY 1)
2
2(1− γ)+ ξ1σY 1 − µY .
The associated optimal asset allocation strategy involves a constant equity weight
p(t,X(t)) = p∗ ≡ σY 1
σ1
+ξ1 − σY 1
(1− γ)σ1
, for all t. (2.2.1)
These results indicate that, although σY 0 does have an impact on the expectedterminal utility, the optimal asset allocation strategy is unaffected by the size ofσY 0. In other words there is nothing that the plan member can do to offset theeffect of unhedgeable volatility in his salary: he just has to accept its existence.At first sight, this conclusion might appear inconsistent with the theory of back-ground risk. Pratt and Zeckhauser (1987), Kimball (1993) and Gollier and Pratt(1996) argue that the introduction of background risk should make risk-averse in-vestors even more risk averse. However, this does not happen here for two reasons.First, when π = 0, the investor’s pension wealth, W (t), is unaffected by the back-ground risk (here, non-hedgeable salary risk) since there are no future premiums.In contrast, the key arguments in background risk theory assume that W (t) isaffected by the background risk. Second, the background risk only appears in theterminal utility function through the numeraire. From equation (2.1.2) we recallthat Y (T ) is the product of its non-hedgeable (background risk) and hedgeablecomponents. Since these are independent and since we are using power utility, wecan separate the expected utility into a product of two expectations. It followsthat the existence of non-hedgeable salary risk (when π = 0) has no impact onthe optimisation problem. We shall see below that when π > 0 the existence ofnon-hedgeable salary risk does result in a lower investment in equities.
The optimal value function for Case 1 is found simply by setting σY 0 = 0 and theoptimal asset allocation strategy is still given by equation (2.2.1).
In both these cases, the fact that the optimal equity weight is unchanging through-out the life of the accumulation programme indicates that lifestyling, whetherdeterministic or stochastic, cannot be the optimal strategy.
7
2.3 Case 3: π > 0, σY 0 = 0
Most DC pension plans involve an ongoing stream of contribution premiums. Thisleads to a significant change in the optimal asset allocation strategy. In Case 3,the future premiums can be fully hedged, that is, a future payment of πY (t) canbe replicated exactly using a combination of cash and equities, as can terminalsalary, Y (T ). In such circumstances, the market is complete and we can attach aunique price to the stream of future premiums. The market price at time t for thepremiums payable between t and T (i.e., their discounted value) can be writtenas
EQ
[∫ T
t
e−r(s−t)πY (s)ds
∣∣∣∣ Ft
]= πY (t)f(t)
where f(t) =exp[(µY − ξ1σY 1)(T − t)]− 1
µY − ξ1σY 1
.
In this equation, Ft is the filtration (or information) generated by Z0(u) and Z1(u)up to time t, and EQ implies expectation with respect to the unique risk-neutralmeasure Q rather than the real-world measure.6
This market price is the key to the solution of Case 3, because it enables us totreat the promised future premiums as if they were part of the current assets ofthe pension plan
W (t) ≡ W (t) + Y (t)πf(t).
We refer to W (t) as the augmented pension wealth (see Boulier et al, 2001, andDeelstra et al, 2000).
Market completeness also allows us to construct a synthetic asset or mutual fund,R2(t), whose dynamics are governed by the SDE
dR2(t) = R2(t) [(r + ξ1σY 1)dt + σY 1dZ1(t)] .
This mutual fund can be used to hedge perfectly the stream of future contributionpremiums, since it is perfectly correlated with salary risk.
Let q(t) represent the proportion of augmented pension wealth, W (t), invested attime t in R1(t), with the remainder invested in R2(t). The holding in R2(t) there-fore comprises (a) a short holding of −πf(t)Y (t) which will be repaid completelyfrom future premiums, and (b) a positive holding equal to (1−q(t))W (t). We canshow, by application of Ito’s formula, that
dW (t) = W (t)[q(t)
((r + ξ1σ1)dt + σ1dZ1(t)
)+ (1− q(t))
((r + ξ1σY 1)dt + σY 1dZ1(t)
)].
6Under Q, dR1(t) = R1(t)[rdt+σ1dZ1(t)] and dY (t) = Y (t)[(r+µY −ξ1σY 1)dt+σY 1dZ1(t)],where Z1(t) is a standard Q-Brownian motion.
8
Now consider X(t) ≡ W (t)/Y (t) = X(t) + πf(t). Note that the terminal utilitycan be expressed in terms of X(T ) as u(X(T )) = γ−1X(T )γ. We can show that
dX(t) = X(t)[(
ξ1σY 1 − µY + q(t)(ξ1 − σY 1)(σ1 − σY 1))dt + q(t)(σ1 − σY 1)dZ1(t)
]and the optimal expected terminal utility is
V (t,X(t)) = h(t)(X(t) + πf(t)
)γ
where h(t) =1
γexp [γθ3(T − t)]
and θ3 =(ξ1 − σY 1)
2
2(1− γ)+ ξ1σY 1 − µY .
It follows that the optimal expected utility differs from Case 1 only because of theinclusion of the present value of the future premiums.
Using V (t,X(t)),the optimal value for q(t) is
q∗(t, X(t)) = q∗ ≡ ξ1 − σY 1
(1− γ)(σ1 − σY 1).
Therefore, the amount of pension wealth invested in equities expressed in units ofY (t) is
X(t)
(q∗ + (1− q∗)
σY 1
σ1
)+ πf(t)
(σ1 − σY 1)
σ1
q∗, (2.3.1)
and the optimal proportion invested in equities is
p∗(t,X(t)) =
(q∗ + (1− q∗)
σY 1
σ1
)+
πf(t)
X(t)
(σ1 − σY 1)
σ1
q∗. (2.3.2)
The relationship between these two time-varying components means that p∗(t,X(t))exhibits what appears to be traditional lifestyle dynamics: that is, it starts high(provided (σ1 − σY 1)q
∗/σ1 > 0) and gradually drifts lower as f(t) decreases andX(t) increases. However, this downward drift is stochastic rather than deter-ministic and falls to q∗ + (1 − q∗)σY 1/σ1 rather than to 0 as under traditionaldeterministic lifestyling. Hence, the optimal asset-allocation strategy can be de-scribed as stochastic lifestyling.
We note that stochastic lifestyling is easy to implement in this case, since theonly time-varying components of p∗(t,X(t)) are (a) the deterministic term f(t)and (b) the stochastic diffusion X(t).
If we employ the optimal asset allocation strategy then
X(T ) = X(T )
= X(0) exp
[(ξ1σY 1 − µY +
(ξ1 − σY 1)2(1− 2γ)
2(1− γ)2
)T +
(ξ1 − σY 1)
(1− γ)Z1(T )
]
9
with X(0) ≡ X(0) + πf(0).
To illustrate, consider the following set of parameters (which are broadly compat-ible with UK data over the last century)7
µY = 0, ξ1 = 0.2, σ1 = 0.2, σY 1 = 0.05, π = 0.1, T = 20 . (2.3.3)
The optimal equity mix is illustrated in Figure 1 which shows the dependenceof the optimal equity proportion and amount, respectively, on X when T − t =20. We can see for low values of X(0) that the optimal equity mix is far fromits asymptotic value of q∗ + (1 − q∗)σY 1/σ1. For completeness, Figure 1 alsoshows the optimal strategy if net wealth, W (t), is allowed to become negative.In practice, negative net wealth might not be an interesting case to consider.However, within the context of the present model (complete market, no constraintson short selling), many strategies can be followed (including the optimal one) thatcould allow net wealth to fall to as low as −πf(t). The completeness of the marketmeans that we can always follow a strategy that will guarantee that the fund willbe positive by time T . In reality most sample paths will be positive for all butthe earliest years of the plan8.
Figure 2 shows p∗(t,X(t)) in two simulations.9 In path B, values are higher be-cause X(t) is lower and close to 0 for the first few years, causing greater volatility.Both paths clearly show the operation of stochastic lifestyling: the optimal equityproportion varies in a stochastic way from year to year, although it will graduallydecline because both t and (in general) X(t) will be increasing. This is why werefer to the optimal strategy as stochastic lifestyling. In contrast, deterministiclifestyling adopts a more risky asset allocation to begin with before shifting in apre-determined way over, say, the final 10 years out of equities into bonds. Fur-ther, with stochastic lifestyling, the equity proportion declines to a non-zero levelwhich depends on both the plan member’s degree of risk aversion and the degreeof correlation with the plan member’s salary. By comparison, the equity propor-tion falls to zero with deterministic lifestyling, irrespective of the plan member’sdegree of risk aversion or salary dynamics (see path C in Figure 2).
7Setting µY = 0 was considered reasonable since long-term average salary increases aresimilar to average long-run interest rates. A reasonable value for r would be 0.06 (nominal),but its value is, in the present model, irrelevant. π can be set to 0.1 without loss of generalitybecause we are using power utility. Any other value could be used for π but there would be noimpact on the optimal strategy or on how it compares with other strategies.
8Just after commencement of the plan, when X = 0, if the fund has gone short in someassets, the Brownian motion will cause sample paths for X(t) to dip slightly below 0 for shortperiods of time.
9The basic shapes of these paths are consistent with the intuition provided by Malkiel (2003)who states that older plan members “have fewer years of labor income ahead of them. Thusthey cannot count on salary income to sustain them should the stock market have a period ofnegative returns. . . . Hence, stocks should comprise a smaller proportion of their assets.” Whatis new here is the stochastic nature of p∗(t,X(t)).
10
−2 0 2 4
−2
−1
01
2
Initial wealth−incomeratio, X(0)
Equ
ity p
ropo
rtio
n
−2 0 2 4
−0.
50.
00.
51.
01.
52.
0Initial wealth−income
ratio, X(0)
Equ
ity a
mou
nt
Figure 1: Left: optimal equity proportion, p∗(0, X(0)). Right: optimal equityamounts, X(0)p∗(0, X(0)), expressed in units of Y (t) when relative risk aversionis 6 (γ = −5) and T = 20. Parameters are µY = 0, ξ1 = 0.2, σ1 = 0.2, σY 1 =0.05, π = 0.1. The asymptotic value for p∗(0, X(0)) as X(0) →∞ is 0.375 (dottedline). The dot on the left indicates the lower limit on X(0).
0 5 10 15 20
01
23
4
A
B
C
Time, t
Equ
ity p
ropo
rtio
n, p
*(t,X
(t))
Figure 2: Optimal equity proportions, p∗(t,X(t)) from two simulated paths (A,B). Parameters as in Figure 1. All simulated paths converge to the dashed line at0.375 as in Figure 1. Deterministic lifestyle strategy (C: dotted line) is shown forcomparison.
11
2.4 Case 4: π > 0, σY 0 6= 0
Qualitatively Case 4 is very different from Case 3. In Case 3 we were able totreat future premiums as a quantifiable part of the current assets of the pensionfund. In the present case, the market is incomplete so that we are unable toborrow explicitly against future premiums. In particular if W (t) is allowed tobecome negative then there is a strictly positive probability that W (T ) will alsobe negative (i.e., the plan becomes insolvent). This forces us to exclude assetallocation strategies that will allow wealth to become negative. This means thatthe optimal strategy for Case 3 is now inadmissible.
We define the general value function J(t, x, p) = E[u(Xp(T ))|X(t) = x], whereXp(t) is the path of X given the asset allocation strategy p = p(t, x). Define P tobe the set of all admissible asset allocation strategies,10 and define
V (t, x) = supp∈P
E[u(Xp(T ))|X(t) = x] = supp∈P
J(t, x, p).
Then V (t, x) satisfies the Hamilton-Jacobi-Bellman equation11
Vt + supp∈P
{µp
XVx +1
2(σp
X)2Vxx
}= 0
subject to V (T, x) =1
γxγ.
In this equation Vt, Vx etc are partial derivatives of V with respect to t, x etc,
µpX = µp
X(t, x)
= π + x(−µY + p(t)σ1(ξ1 − σY 1) + σ2
Y 0 + σ2Y 1
)and (σp
X)2 = σpX (t, x)2
= x2(σ2
Y 0 + (p(t)σ1 − σY 1)2).
For a given (t, x), we now solve the static supremum problem which results in
p∗(t, x) ≡ p∗(t, x; V ) =1
σ1
(σY 1 − Vx
xVxx
(ξ1 − σY 1)
). (2.4.1)
The optimal solution for p thus depends on the optimal value function V (t, x),
10For a discussion of admissible strategies see, for example, Korn (1997).11In the complete market cases (1 and 3) we could use the alternative martingale approach
of Karatzas et al (1987) and Cox & Huang (1991). This can offer a more direct route to thesolution for Cases 1 and 3 but is much less easy to apply in the incomplete market case (Cases2 and 4). For consistency, then, we focus on the HJB approach.
12
which is found by solving numerically the PDE
Vt + µp∗X Vx +
1
2
(σp∗
X
)2
Vxx = 0 (2.4.2)
subject to V (T, x) =1
γxγ
where µp∗X = π + x
(−µY + p∗(t, x; V )σ1(ξ1 − σY 1) + σ2Y 0 + σ2
Y 1
)and
(σp∗
X
)2
= x2(σ2
Y 0 +{p∗(t, x; V )σ1 − σY 1
}2)
.
(Recall that µp∗X and σp∗
X are the drift and volatility of X(t) given strategy p∗.)
Remark 2.4.1
Equation (2.4.1) shows that the optimal risky portfolio p∗(t, x) can be expressed asa combination of two fixed portfolios, σY 1/σ1 and (ξ1 − σY 1)/σ1, with the balancebetween the two depending on t and x only through V (t, x).
Figure 3 presents numerical results for γ = −5 (RRA = 6) and the followingparameter set:
µY = 0, ξ1 = 0.2, σ1 = 0.2, σY 1 = 0.05, σY 0 = 0.05, π = 0.1, T = 20. (2.4.3)
In each plot, we give the optimal value function (solid lines) for the terminal utilityfunction V (t, x) for T − t = 10 and 20 years to maturity. We note from this plotthat V (t, x) (for t < T ) has a finite limit as x → 0+. (In contrast, V (T, x) → −∞as x → 0+.) This happens because when X(t) = 0, we will always have X(T ) > 0due to future premiums and the finite limit for V (t, x) as x → 0+ indicates thatX(T ) is not concentrated too close to 0.12
Figure 3 also plots the suboptimal value function (dashed line) J(t, x, p) whenp = 0.375 for all (t, x) (the limiting value for p∗(t, x) as x → ∞). This figuresuggests that the differences between the optimal and suboptimal value functionsare not too great and diminish as x increases. However, a comparison of expectedterminal utilities only tells us that one strategy is better than another and doesnot allow us to quantify the cost of suboptimality to the plan member. This is anissue that we will come back to in Section 3.4.1.
The corresponding optimal dynamic asset allocation strategies p∗(t, x) are plottedin Figure 4 for T − t = 0, 10 and 20 years to maturity. We note that p∗(t, x) →∞as x → 0+ (top graph). However, when we look at the amount invested, weobserve (particularly from the bottom graph) that p∗(t, x)x converges to 0 at thesame rate as
√x.13 This is in clear contrast to Case 3 where limx→0 p(t, x)x > 0.
12Equivalently, the left-hand tail of the distribution of logX(T )|X(t) = 0 is not too dense.13That is, (p∗(t, x)x)/
√x = p∗(t, x)
√x → c, for some constant c ≥ 0, as x → 0.
13
The behaviour of p∗(t, x)x close to x means that in Case 4 X(t) can never becomenegative (in contrast to Case 3).14 It follows that the Case 4 asset allocationstrategy is rather more conservative than that of Case 3 when the fund size is low;this is to avoid the risk of bankrupting the fund.
Individual realisations of p∗(t, x) against t would look similar to the stochasticlifestyling in Figure 2. However, the square-root convergence property meansthat the sample paths have slightly lower equity weightings on average. Thisrelationship is consistent with the theory of background risk discussed in Section2.2.
3 A more general stochastic model
3.1 The structure of the model
We will now incorporate three extensions to the problem:
• the introduction of a stochastic risk-free nominal rate of interest, r(t);
• the extension of the investment opportunity set to N risky assets ratherthan 1;
• the introduction of the replacement ratio15 as an argument in the terminalutility function.
The components of the model are as follows:
• The risk-free rate of interest is a one-factor diffusion process governed bythe time-homogeneous SDE
dr(t) = µr(r(t))dt +N∑
j=1
σrj(r(t))dZj(t)
where the Zi(t) are independent, standard Brownian motions. We defineσr(r) = (σr1(r), . . . , σrN(r))′. The value of units in the cash fund at t is
then R0(t) = R0(0) exp[∫ t
0r(s)ds
].
14In fact, square root convergence is a necessary condition for X(t) to avoid becoming negative(see, for example, Duffie et al., 1997). If limx→0 p∗(t, x)
√x = ∞ then X(t) could become
negative.15The ratio of the initial pension at retirement to the final salary before retirement.
14
0.0 0.5 1.0 1.5 2.0
−0.
20−
0.15
−0.
10−
0.05
0.00
Wealth−income ratio, X(10) = x
Exp
ecte
d te
rmin
al u
tility
t=10, T=20
0.0 1.0 2.0 3.0 4.0
−0.
004
−0.
003
−0.
002
−0.
001
0.00
0
Wealth−income ratio, X(0) = x
Exp
ecte
d te
rmin
al u
tility
t=0, T=20
Figure 3: Value functions for 10 and 20 years to maturity when relative riskaversion is 6 (γ = −5) and π = 0.1. Solid lines show the optimal V (t, x), whiledashed lines show the suboptimal value function J(t, x, p) where p = 0.375 for allt, x. Both value functions are calculated using the finite difference method.
15
0.0 0.5 1.0 1.5 2.0
01
23
4
Wealth−income ratio, X(t)
Opt
imal
equ
ity p
ropo
rtio
nt=0, T=20t=10, T=20t=20, T=20
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.6
1.0
A
Wealth−income ratio, X(t)
Opt
imal
equ
ity a
mou
nt
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.1
0.2
0.3
Wealth−income ratio, X(t)
Opt
imal
equ
ity a
mou
nt
A
Figure 4: Optimal asset-allocation strategies when relative risk aversion is 6 (γ =−5), π = 0.1 and T = 20. Top: equity proportions p∗(t, x) for t = 0 (solid line),t = 10 (dashed) and t = 20 (dotted). Middle: amounts invested in equities,p∗(t, x)x, for t = 0 (solid line), t = 10 (dashed) and t = 20 (dotted). Bottom: boxA in the middle graph magnified.
16
• There are N risky assets. Let Ri(t) be the total return16 on an investmentsolely in asset i with
dRi(t) = Ri(t)
[(r(t) +
N∑j=1
σijξj
)dt +
N∑j=1
σijdZj(t)
]
for i = 1, . . . , N . The volatility matrix C = (σij)Ni,j=1 is assumed to be
constant as are the market prices of risk, the ξ = (ξ1, . . . , ξN)′. The riskpremium on asset i is
∑Nj=1 σijξj.
• The plan member’s salary, Y (t), evolves according to the SDE
dY (t) = Y (t)
[(r(t) + µY (t)) dt + σY 0dZ0(t) +
N∑j=1
σY jdZj(t)
]
where µY (t) is a deterministic function of time, the σY j’s are constants andZ0(t) is a standard Brownian motion, independent of Z1(t), . . . , ZN(t). Wedefine the vector σY = (σY 1, . . . , σY N)′.
• The value of the plan member’s pension fund (pension wealth) is denotedby W (t) and has the SDE
dW (t) = W (t) [(r(t) + p(t)′Cξ) dt + p(t)′CdZ(t)] + πY (t)dt
where Z(t) = (Z1(t), . . . , ZN(t))′ and p(t) = (p1(t), . . . , pN(t))′ are the pro-portions of the fund invested in the various risky assets.
In this problem, the control variable is p(t), which we will initially take tobe unconstrained. The set P will be used to denote the set of all admissiblecontrols, p(t).
• At the time of retirement at age ζ, the fund is used to purchase a pensionat the prevailing market rate for life annuities, a(T, r(T )). For example, fora level annuity of 1 unit per annum payable continuously we have (for ageneral retirement date t)
a(t, r(t)) =
∫ ∞
0
b(t, t + s; r(t))φζ(t, s)ds. (3.1.1)
Here
– b(t, τ ; r) is the price at time t of a zero-coupon bond maturing at timeτ when the risk-free rate of interest at time t is r.
16That is, the value of a single premium investment in asset i with reinvestment of dividendincome.
17
– φζ(t, s) is the probability of survival from time t to t + s for a life agedζ at time t. We assume that φζ(t, s) is known at time 0.
The rate per annum of the continuously-paid pension purchased at T is
P (T ) =W (T )
a(T, r(T )).
• We assume the plan member’s terminal utility depends on both his finalsalary and his pension wealth at retirement. We will focus on two specialcases: (a) the ratio of terminal pension wealth to final salary, X(T ) =W (T )/Y (T ), and (b) the replacement ratio, H(T ) = P (T )/Y (T ) =X(T )/a(T, r(T )). Thus terminal utility will be of the form
u(X(T ), r(T )
) ≡ u(X(T )
)or u
(X(T )/a(T, r(T ))
).
If utility is some function of P (T )/Y (T )17 the effect of habit formation ismore apparent (in the sense proposed by Ryder and Heal, 1973, Sundaresan,1989, and Constantinides, 1990), since we are comparing consumption afterretirement with the rate of consumption just before retirement.
Expected terminal utility is then given by
J(t, x, r; p) = E[u(Xp(T ), r(T )
) | X(t) = x, r(t) = r]
where Xp(t) is the path of X(t) given the strategy p.
Our aims are twofold. The first is to determine the plan member’s optimal ex-pected terminal utility, that is, to find
V (t, x, r) = supp∈P
J(t, x, r; p), (3.1.2)
and to determine the strategy p that attains this maximum. In addition, we wishto evaluate the performance of a variety of popular asset allocation strategiesrelative to this theoretical benchmark.
Given the wealth to salary ratio, X(t) = W (t)/Y (t), a straightforward applicationof the product rule gives us the SDE
dX(t) = X(t)[(− µY (t) + p(t)′C(ξ − σY ) + σ2
Y 0 + σ′Y σY
)dt
−σY 0dZ0(t) +(p(t)′C − σ′Y
)dZ(t)
]+ πdt.
17More precisely, we could look at P (T )/(1 − π)Y (T ) since (1 − π)Y (T ) and P (T ) are theconsumption rates just before and just after retirement. However, it is not necessary for us toinclude explicitly the (1 − π) factor. First, π is an exogenously specified parameter. Second,since we are using power utility the (1− π) factor results in a constant multiplier in the utilitythat has no impact on the optimisation problem. For these reasons we prefer to leave out the(1− π).
18
3.2 The Hamilton-Jacobi-Bellman equation
Our approach to solving the optimisation problem (equation 3.1.2) is to use theHamilton-Jacobi-Bellman (HJB) equation. The HJB equation for this problemis (see, for example, Merton, 1969, 1971, 1990, Korn, 1997, Øksendal, 1998, orBjork, 1998)
Vt + supp∈P
{ApV}
= 0, (3.2.1)
where
Ap = µr(r)∂
∂r+ µp
X
∂
∂x+
1
2νrr
∂2
∂r2+ νp
rx
∂2
∂r∂x+
1
2νp
xx
∂2
∂x2
µpX = x
(− µY (t) + p′C(ξ − σY ))
+ π
µY (t) = µY (t)− σ2Y 0 − σ′Y σY
νrr = σr(r)′σr(r)
νprx =
(p′C − σ′Y
)σr(r)x
and νpxx =
(σ2
Y 0 +(p′C − σ′Y
)(C ′p− σY
))x2.
If we follow the usual steps (see, for example, Bjork, 1998) we find that the optimalasset allocation strategy takes the form
p∗(t, x, r; V ) = C ′−1
(σY − (ξ − σY )
Vx
xVxx
− σr(r)Vxr
xVxx
). (3.2.2)
If we now insert this expression for p∗(t, x, r; V ) into equation (3.2.1) and simplifywe get the PDE for V (t, x, r)
Vt + µr(r)Vr +(π − µY (t)x + σ′Y (ξ − σY )x
)Vx +
1
2σr(r)
′σr(r)Vrr +1
2σ2
Y 0x2Vxx
−1
2(ξ − σY )′(ξ − σY )
V 2x
Vxx
− (ξ − σY )′σr(r)VxVxr
Vxx
− 1
2σr(r)
′σr(r)V 2
xr
Vxx
= 0.
(3.2.3)
Before we go on to look at the solution of the PDE (3.2.3) we will make some ob-servations about the composition of the optimal portfolio p∗(t, x, r; V ) in equation(3.2.2).
Theorem 3.2.1 [Three-fund theorem]
The optimal asset mix, p∗(t, x, r; V ) (equation 3.2.2), at any given time, consists
19
of investments in three efficient mutual funds as follows:
p∗(t, x, r; V ) = θApA + θBpB + θCpC (3.2.4)
where θA = θA(t, x, r) = 1− Vxr − da(r)Vx
da(r)xVxx
θB = θB(t, x, r) =Vxr
da(r)xVxx
and θC = θC(t, x, r) = 1− θA(t, x, r)− θB(t, x, r) = − Vx
xVxx
with pA = C ′−1σY
pB = C ′−1(σY − da(r)σr(r)
)and pC = C ′−1
ξ. (3.2.5)
Proof: See Appendix A.
We interpret (3.2.4) and (3.2.5) as follows. The optimal weight in risky assets isequivalent to investing in three efficient mutual funds denoted A, B and C. Thethree mutual funds can be interpreted as follows:
A: Fund A is the minimum-risk portfolio measured relative to the salary nu-meraire, Y (t), and its purpose is to hedge salary risk. Asset proportionsare given by the vector pA (with pA0 = 1 −∑N
i=1 pAi). Since this fund willbe dominated by cash (and contain 100% cash if salary growth and assetreturns are uncorrelated, but also contain other assets if salary growth iscorrelated with their returns), we will refer to Fund A for convenience asthe ‘cash’ fund.
B: Fund B is the minimum-risk portfolio measured relative to Y (t)/a(t, r(t))and its purpose is to hedge annuity risk. Asset proportions are given by thevector pB (with pB0 = 1−∑N
i=1 pBi). Since this fund will be dominated bybonds whose returns are highly correlated with annuity yields, we will referto Fund B as the ‘bond’ fund.
C: Fund C is a risky portfolio which is efficient when measured relative to bothY (t) and Y (t)/a(t, r(t)). Asset proportions are given by the vector pC (withpC0 = 1−∑N
i=1 pCi). The fund will be dominated by equities and so we willrefer to it as the ‘equity’ fund: its purpose is to satisfy the risk appetite ofthe plan member.
Mutual funds A and C (equation 3.2.5) maintain constant proportions in each ofthe N + 1 assets. The equivalent proportions in fund B will vary over time, but
20
only in response to changes in r(t) rather than, separately, to changes in t or X(t).However, the proportions of the overall pension fund invested in each of the threemutual funds (that is, θA(t, x, r), θB(t, x, r), θC(t, x, r) ) depend on all of t,X(t)and r(t), according to equations (3.2.5). As in Section 2 the stochastic nature ofthe paths of θA(t, x, r), θB(t, x, r) and θC(t, x, r) over time means that the optimalasset allocation strategy can usefully be described as stochastic lifestyling.
Remark 3.2.2
The term θC is equal to the reciprocal of the degree of relative risk aversion.
It follows that, since relative risk aversion is positive (but possibly dependent upont and x), then the investment in portfolio C is necessarily positive. In addition, ifrelative risk aversion is constant, it will be optimal to invest a constant proportionin the risky portfolio C ′−1ξ over time.
Remark 3.2.3
The three portfolios, pA, pB and pC, do not depend upon the level of non-hedgeablesalary risk, σY 0. However, the precise mix18 will depend upon σY 0 through its effecton V (t, x, r).
Provided that all members have the same values for σY , the same three fundscan be used for all plan members no matter what their idiosyncratic risk (σY 0),age, wealth or attitude to risk. This has the important practical consequencethat pension providers can use these three general funds to satisfy the needs ofmany plan members instead of having to provide tailor-made portfolios for eachindividual.
Corollary 3.2.4
Suppose that V (T, x, r) = K (x/a(T, r)), that is, the terminal utility is a functionof the pension as a proportion of final salary (replacement ratio) achieved at timeT . Then θA(T, x, r) = 0 for all x, r.
Proof:
18We will see how this mix varies stochastically in the numerical example later in this section.
21
At t = T, we find that:
Vx =1
a(T, r)K ′(x/a(T, r))
Vxr =da(r)
a(T, r)K ′(x/a(T, r)) +
xda(r)
a(T, r)2K ′′(x/a(T, r))
Vxx =1
a(T, r)2K ′′(x/a(T, r)).
It is then straightforward to confirm that θA(T, x, r) = 0.
¤This result shows what happens to the asset allocation when the plan member isconcerned about receiving a pension at retirement, rather than a cash lump sum.Although part of the pension wealth will generally be invested in fund A prior toretirement, as the retirement date approaches, the weight in fund A is reduced tozero. The exception to this result occurs if the pension plan is funding for a cashlump sum at retirement rather than a pension. In this case da(r) = 0 for all r andportfolios A and B are identical.
Conjecture 3.2.5
As T − t tends to infinity θB(t, x, r) tends to zero.
We make this conjecture on the following basis. The further we are from retirementthe less able are we to predict what interest rates will be at the time of retirement.This means that Vr and Vxr are likely to tend to 0 as T −t increases. For a specificexample where the conjecture is true, see Section 3.4.1 and Equation (3.4.7).
3.2.1 Boundary conditions for the PDE
So far we have derived general results which do not depend upon a specific form foru(x, r). However, if we wish to obtain further results, we need to be more specificabout the terminal utility, u(x, r). This will give us the boundary condition forthe PDE (equation 3.2.3: V (T, x, r) = u(x, r)). In the sections that follow, we wewill narrow our analysis to terminal utilities which are power utilities in x, whilekeeping a more general functional form in r: that is,
u(x, r) =1
γg(T, r)1−γxγ
for some general function g(T, r). A full power utility function will be employedin the numerical example in Section 3.4.1.
We are now in a position to discuss two of the four cases analysed previously inSection 2. Case 2 (π = 0, σY 0 6= 0) and Case 4 (π > 0, σY 0 6= 0) involve a type
22
of computational analysis that is sufficiently different and sufficiently extended tojustify a separate paper.19
3.3 Case 1: π = 0, σY 0 = 0
Let us first consider the solution for (3.2.3) for the single-premium Case 1 whereπ = 0 and σY 0 = 0. While this case is not particularly interesting in itself, it leadsus directly to the solution for Case 3 where π > 0 and σY 0 = 0.
Theorem 3.3.1
V (t, x, r) is of the form γ−1g(t, r)1−γxγ where g(t, r) satisfies the PDE
gt +1
2σr(r)
′σr(r)grr +
(µr(r) +
γ
(1− γ)(ξ − σY )′σr(r)
)gr
+γ
(1− γ)
[− µY (t) + σ′Y ξ − 1
2(γ − 1)(ξ − σY )′(ξ − σY )
]g = 0.
(3.3.1)
The boundary condition for g(T, r) is defined as {γV (T, x, r)x−γ}1/(1−γ) or{γu(x, r)x−γ}1/(1−γ).
Proof: See Appendix.
Corollary 3.3.2
By the Feynman-Kac formula (see, for example, Bjork, 1998) there exists a prob-ability measure20 Q(γ) such that
g(t, r(t)) = EQ(γ) [g(T, r(T ))D(t, T ) | Ft]
where r(s) is governed by the SDE
dr(s) = µr(r(s))ds + |σr(r(s))|dZ(s)
µr(r) = µr(r) +γ
1− γ(ξ − σY )′σr(r)
r(t) = r(t)
19However, we conjecture that the outcomes would be a combination of what we have alreadyobserved in both the deterministic and stochastic r(t) models.
20The measure Q(γ) is (like the risk-neutral measure in derivative pricing) an artificial prob-ability measure which provides us with a convenient computational tool. It does not imply thatinvestors with different levels of risk aversion use different probabilities.
23
where |σr(r(s))| is the modulus of σr(r(s)), Z(s) is a standard, one-dimensionalBrownian motion under the artificial measure Q(γ),
D(t, T ) = exp
[γ
1− γ
∫ T
t
θ(s)ds
]
θ(s) = −µY (s) + σ′Y ξ − 1
2(γ − 1)(ξ − σY )′(ξ − σY )
⇒ D(t, T ) = exp
[γ
1− γ
{−MY (t, T ) + ψ(γ)(T − t)
}]
MY (t, T ) =
∫ T
t
µY (s)ds
and ψ(γ) = σ′Y ξ − 1
2(γ − 1)(ξ − σY )′(ξ − σY ). (3.3.2)
Consider the optimal asset mix. The general form for V (t, x, r) reveals immedi-ately that the proportion of the fund invested in portfolio pC is constant: that is,θC(t, x, r) = 1/(1 − γ) for all (t, x, r). The proportions θA and θB invested in pA
and pB, respectively, will depend upon t and r but not upon x. In particular, wenote that since EQ(γ)[g(T, r(T ))|Ft] depends upon the particular model chosen forr(t), so will θA(t, x, r) and θB(t, x, r).
Suppose also that r(t) is stationary and ergodic under Q(γ) and thatEQ(γ)[g(T, r(T ))|Ft] → constant and is finite as T − t → ∞. Then, as T − t →∞, g(T, r(t))/D(t, T ) → constant for all r(t). It follows that the proportion,θB(t, x, r), invested in pB tends to 0 as T − t →∞ (Conjecture 3.2.5).
On the other hand, if the plan is funding for a cash lump sum at T , rather than apension, then g(T, r) ≡ 1 and θB(t, x, r) ≡ 0 for all (t, x, r) (see equation (3.2.5)).
3.4 Case 3: π > 0, σY 0 = 0
Since salaries are hedgeable, we can attach a unique price to future pension con-tributions. Let Q be the risk-neutral pricing measure under which the N riskyassets have the dynamics
dRi(t) = Ri(t)
[r(t)dt +
N∑j=1
σijdZj(t)
]
24
and where the Zj(t) are independent standard Q-Brownian motions. Also we haveunder Q
dY (t) = Y (t)
[(r(t) + µY (t)−
N∑j=1
σY jξj
)dt +
N∑j=1
σY jdZj(t)
]
⇒ Y (τ) = Y (t) exp
{ ∫ τ
t
(r(s) + µY (s))ds−N∑
j=1
σY j
(ξj +
1
2σY j
)(τ − t)
+N∑
j=1
σY j
(Zj(τ)− Zj(t)
) }.
The market value at time t for future premiums payable between t and T is then
EQ
[∫ T
texp
{−
∫ τ
tr(s)ds
}πY (τ)dτ
∣∣∣∣ Ft
]
= πEQ
[ ∫ T
tY (t) exp
{∫ τ
tµY (s)ds− σ′Y ξ(τ − t) −1
2|σY |2(τ − t) + σ′Y
(Z(τ)− Z(t)
) }dτ
∣∣∣∣∣ Ft
]
= πY (t)∫ T
texp
{MY (t, τ)− σ′Y ξ(τ − t)
}dτ (by 3.3.2)
= πY (t)f(t), say. (3.4.1)
Theorem 3.4.1
V (t, x, r) is of the form γ−1g(t, r)1−γ(x + πf(t))γ where g(t, r) satisfies the PDE(3.3.1) with boundary condition g(T, r) = γV (T, x, r)x−γ.
Proof: We only need to note that the optimal strategy is composed of two parts.At time t:
A: hold −πY (t)f(t) in the replicating portfolio which will be paid off exactlyby future premiums;
B: invest the surplus W (t) = W (t) + πY (t)f(t) in portfolios pA, pB and pC inthe same proportions θA, θB and θC as in Case 1 where π = 0 and σY 0 = 0.
This will produce the same expected terminal utility as the single premium Case1. Any other strategy will generate a lower value.
¤
25
3.4.1 A numerical example using the Vasicek model
Let us now illustrate how the optimal asset allocation strategy varies with (t, x, r)for a specific example. In this example we will assume that plan members con-tribute to their pension plan at the rate of 10% of salary per annum: that is,π = 0.1.
We will now use the Vasicek (1977) model for r(t), which will allow us to derivetractable results. Further, bearing in mind the three-fund theorem above, we canrestrict the asset allocation to three mutual funds, a cash fund, a bond fund andan equity fund. Thus we will take
dr(t) = αr(µr − r(t))dt + σr1dZ1(t)
with σr2 = 0. The parameter values in our model will be
αr = 0.25, µr = 0.06, σr1 = −0.02
C =
(0.1 00.1 0.2
), ξ =
(0.20.3
), σY =
(0.020.02
). (3.4.2)
From the structure of C and σr we see that R1(t) is the bond fund and that R2(t)is the equity fund.
For simplicity, we will assume that a(t, r(t)) is of the form exp[d0 − d1r(t)]. Thiswill keep things tractable without seriously altering the qualitative observationsin our example. We will assume that d0 = 3 and d1 = 3.5 (which implies thata(t, r(t)) behaves like a zero-coupon bond with 8.318 years to maturity).
It follows that
pA = C ′−1σY =
(0.10.1
), pC = C ′−1ξ =
(0.51.5
),
and pB = C ′−1(σY − d1σr) =
(0.80.1
). (3.4.3)
Now, under the artificial measure Q(γ)
dr(s) = µr(r(s))ds + σr1dZ(s)
where µr(r) = αr(µr − r)
and µr = µr +γ(ξ1 − σY 1)σr1
(1− γ)αr
= 0.06− 0.0144γ
1− γ.
We will now take the terminal utility function
u(x, r) = V (T, x, r) =1
γ
(x
a(T, r)
)γ
=1
γe−γd0+γd1rxγ
⇒ g(T, r) = exp
[γ
1− γ(−d0 + d1r)
]. (3.4.4)
26
We will make the final parameter choices as follows:
γ = −5, µY ≡ 0.
Having defined g(T, r) we can now use Corollary 3.3.2 to derive a functional formfor g(t, r). The various quantities defined in equation (3.3.2) are ψ(γ) = 0.1092333(as required for D(t, T )), µr = 0.072, and MY (t, T ) ≡ 0. Under Q(γ), r(T ), givenr(t), is normally distributed with
EQ(γ) [r(T )|r(t)] = µr + (r(t)− µr)e−αr(T−t)
and V arQ(γ) [r(T )|r(t)] = σ2r1
(1− e−2αr(T−t)
)2αr
⇒ EQ(γ) [g(T, r(T )) | r(t)] = exp
[A(γ, T − t) + B(γ, T − t)r(t)
1− γ
]
where A(γ, τ) = −γd0 + γd1µr
(1− e−αrτ
)+
1
2
γ2d21σ
2r1
(1− γ)
(1− e−2αrτ )
2αr
and B(γ, τ) = γd1e−αrτ .
Hence g(t, r(t))1−γ = exp [A(γ, T − t) + B(γ, T − t)r(t) + γψ(γ)(T − t)] .
(3.4.5)
This implies that
V (t, x, r) =1
γeA(γ,T−t)+γψ(γ)(T−t)eB(γ,T−t)r
(x + πf(t)
)γ(3.4.6)
⇒ θA(t, x, r) = −πf(t)
x+
(x + πf(t)
)x
γ
(γ − 1)
(1− e−αr(T−t)
)θB(t, x, r) =
(x + πf(t)
)x
γe−αr(T−t)
(γ − 1)
and θC(t, x, r) =
(x + πf(t)
)x
1
(1− γ). (3.4.7)
These equations give us explicit formulae for the stochastic lifestyle strategy. In(3.4.7), θA(t, x, r) has been written in a way which highlights the two componentsto investments in pA. First, we have a short holding of −πf(t) which will be paidoff precisely by the future premiums (since we have a complete market). Second,we have the augmented wealth X(t)+πf(t) which is invested in fixed proportionsin pA, pB and pC which vary with the term to maturity only21 22.
21We note that none of the portfolio weights θA(t, x, r), θB(t, x, r) and θC(t, x, r) depends uponr. This is a consequence of the choice of the Vasicek model and the simple form for a(t, r(t)).In other cases the θ’s will depend upon r as well as on t.
22We can note here a similarity to the concept of portfolio insurance (see, for example, Blackand Jones, 1988, Black and Perold, 1992, and Cairns, 2000). Portfolio insurance is an investment
27
To complete the numerical example we note from equation (3.4.1) that:
f(t) =
∫ T
t
exp {−σ′Y ξ(s− t)} ds =
∫ T
t
exp[−0.01(s− t)]ds
=1− e−0.01(T−t)
0.01.
Examples of five scenarios are plotted in Figure 5 for this parameter set, togetherwith π = 0.1 and γ = −5 (RRA = 6). The top graph in this figure shows thewealth-income ratio, X(t), while the middle graph shows the replacement ratiothat could be achieved with the current fund, current income and current annuityrates (that is, X(t)/a(t, r(t))). The five paths in the top two graphs give us anindication of the general spread of results. The minimum admissible fund sizefor the avoidance of insolvency (−πf(t)) is also plotted for reference and we cansee that the actual fund size is always comfortably above this at all times on allsample paths. Compared with the top graph, the five paths in the middle graphare less spread out and smoother as the maturity date approaches because annuityrisk is being hedged.
This observation is clearer in Figure 6 where we consider an extremely risk averseplan member. Since the market is complete in this case, the plan member isboth willing and able to target a specific replacement ratio with certainty. Thismeans that in the middle graph the sample replacement ratios, X(t)/a(t, r(t)), allconverge to the same point at T = 20. At intermediate times, t, they are morespread out because the investment strategy is targeting for certainty at T = 20rather than t < T . This plot demonstrates the importance of ‘seeing the strategythrough to its conclusion’. In the top graph in Figure 6 the tightness of the samplepaths up to t = 10 reflects the dominance of Fund A in the asset allocation strategy(as shown in the bottom graph). It is only when the strategy switches to Fund Bthat X(t) starts to show significant variability. This is because Fund B is riskyrelative to the salary numeraire, Y (t). In contrast, when risk aversion is low, thenX(t)/a(t, r(t)) does not converge as t → T (Figure 5 middle graph).
In the bottom graph in Figure 5 we have selected the bold scenario in the uppergraphs in order to show how the asset allocation strategy varies over time. Thestochastic-lifestyling nature of the strategy is evident. Initially, when X(t) is smallthere is considerable short-selling of the cash fund A (in other words borrowingcash) in anticipation of future premiums. Also for small t the asset mix shows afair degree of randomness (which is different for each of the 5 scenarios). This is
strategy that aims to ensure that the value of a pension plan never falls below a specified floor.Here the floor of 0 at time T imposes a floor of −πf(t) at time t. Furthermore, the form ofthe utility function at T dictates the way that we invest the surplus over the floor. Since thevalue of the floor is negative, in contrast to the traditional positive floor in portfolio insurance,greater net pension wealth implies a lower proportion invested in equities.
28
because the asset mix is most sensitive to changes in x when x is small, whichwill usually be when t is also small. Later on the asset mix seems to follow arelatively smooth path, when X(t) is larger. For clarity, only one sample path hasbeen plotted in the bottom graph. When we plot several sample paths, say, forθC(t) we would find that the individual paths can be quite different early on inthe contract23. However, the sample paths θC(t) all converge to the same limitingvalue.
3.4.2 The cost of suboptimality
Up until now we have focused on the derivation and analysis of the optimalstochastic asset-allocation strategy. So we have worked our way through thetheory. Now we have to ask ourselves: is it worth our whiles trying to implementsuch a strategy in practice? Rephrasing this question: does the use of the optimalstochastic lifestyle strategy significantly improve the welfare of a plan memberrelative to the more commonly used DC strategies? If it turns out that stochasticlifestyling does not improve welfare significantly then the effort of trying to imple-ment it might not be worthwhile. In contrast, if we find that stochastic lifestylingdoes add significant value then we should be trying to persuade pension providersto consider alternative strategies that are closer to stochastic lifestyling than thecurrent range of strategies that are typically offered.
In Table 1 we give numerical results for different levels of risk aversion (RRA =1, 6, 12 corresponding to 24 γ = 0, − 5, − 11) and policy durations (T = 20, 40years). For each (γ, T ) combination we have considered seven asset allocationstrategies:
• Optimal stochastic lifestyle strategy, p∗(t,X(t), r(t)).
• Salary-hedged static strategy (S). This is the strategy
p(t,X(t), r(t)) =−γ
1− γpB +
γ
1− γpC for all (t,X(t), r(t)).
This is the limiting value of p∗(t,X(t), r(t)) as t → T and takes account ofthe correlation between salaries and the risky assets as well as the conversionof pension wealth into an annuity at T .
• Merton-static strategy (M). This is the static strategy
p(t,X(t), r(t)) =γ
1− γpC for all (t,X(t), r(t)).
23For example, for the 5 sample paths plotted in Figure 5 (top and middle) θC(5) ranged from0.44 to 0.78.
24We actually set γ = +0.01 which gives essentially the same results as for γ = 0, but iscomputationally more convenient.
29
0 5 10 15 20
−2
02
4
Time, t
X(t
)
− πf(t)
0 5 10 15 20
−10
010
2030
Time, t
X(t
)/a(
t,r(t
)) (
%)
0 5 10 15 20
−1.
00.
01.
02.
0
θA(t)
θB(t)
θC(t)
Time, t
Pro
port
ion
of fu
nd
Figure 5: Dynamics of a plan member’s pension wealth over 0 < t < T = 20 forrelative risk aversion of 6 (γ = −5), and π = 0.1. Top: X(t) = W (t)/Y (t);5 scenarios (solid lines) with the minimum −πf(t) (dashed line). Middle:X(t)/a(t, r(t)) = W (t)/Y (t)a(t, r(t)); same 5 scenarios (solid lines). Bottom:proportions invested in portfolios pA, pB and pC : θA(t) (solid line), θB(t) (dashedline) and θC(t) (dotted line) corresponding to the bold scenario in the top andmiddle graphs.
30
0 5 10 15 20
−0.
50
1.0
2.0
Time, t
X(t
)
− πf(t)
0 5 10 15 20
05
1015
Time, t
X(t
)/a(
t,r(t
)) (
%)
0 5 10 15 20
−1.
00.
01.
02.
0
θA(t)
θB(t)θC(t)
Time, t
Pro
port
ion
of fu
nd
Figure 6: Dynamics of a plan member’s pension wealth over 0 < t < T = 20 forrelative risk aversion 1001 (γ = −1000), and π = 0.1. Top: X(t) = W (t)/Y (t);5 scenarios (solid lines) with the minimum −πf(t) (dashed line). Middle:X(t)/a(t, r(t)) = W (t)/Y (t)a(t, r(t)); same 5 scenarios (solid lines). Bottom:proportions invested in portfolios pA, pB and pC : θA(t) (solid line), θB(t) (dashedline) and θC(t) (dotted line) corresponding to the bold scenario in the top andmiddle graphs.
31
This is the classical Merton strategy which does take account of relative riskaversion, RRA = 1−γ, but does not make allowance for correlation betweensalary and the asset returns, or the pension conversion.
• Deterministic lifestyle strategy. The general lifestyle strategy invests100% in equities (Fund C) until τ = 10 or 5 years before retirement. Overthe final τ , years the equity investments are gradually switched wholly intobonds (Fund B) or wholly into cash (Fund A). The four strategies are la-belled B for a switch into Fund B or A for Fund A; and by the length of theswitching period (5 or 10 years).
In each sub-table ((a) to (f)) there are two rows of numbers. In the first row,we give the expected terminal utility. For convenience these have been rescaledso that the optimal value is +100 or -100 (depending on whether the raw valueis positive or negative). This row only allows us to rank the different strategies.The numbers give us no indication of how much of an improvement in welfare aplan member will get as a result of using the optimal stochastic strategy. In thesecond row, therefore, we give a more concrete measure of the cost of adoptingthe suboptimal strategy relative to the optimal one. For example, in Table 1(a) the static strategy S has a cost of 37.9% of the original contribution rate of10% of salary: that is, the plan member would have to pay a contribution rate ofπ′ = π × 1.379 or 13.79% of salary to get the same expected terminal utility asthe theoretical optimum.25
We can make the following observations:
• The cost is higher for 40-year time horizons than for 20-year horizons. Thisreflects various interacting factors. However, the main reason is simply thatthe longer the duration of suboptimal investment the greater the the costsof suboptimality.
• The costs of suboptimality vary substantially as we move from one level ofrisk aversion to another but often vary in a non-monotonic way. For ex-ample, strategy B-5 with T = 40 has costs of 196.7%, 40.4% and 97.2%of the original contribution rate for members with RRA’s of 1, 6 and 12respectively. There are two reasons for this variation with the RRA. First,the deterministic strategy might give a reasonably good approximation top∗(t, x) for certain values (e.g. medium) of γ. The more accurate the approx-imation the lower the cost of suboptimality. (For example, static strategy S
25To place this on a more theoretical foundation: the plan sponsor would have to increasethis member’s salary from Y (t) to Y (t) = 1.0379Y (t) and the contribution rate would be π =0.1379/1.0379 or 13.29% of the revised salary, Y (t). Under this revised scheme the pre-retirementconsumption rate (1− π)Y (t) = (1− π)Y (t) is unaltered.
32
(a) γ = 0 (RRA = 1), T = 20Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) 100 99.68 99.68 99.30 99.38 99.24 99.35Cost 0% 37.9% 37.8% 101.8% 86.7% 113.9% 92.3%
(b) γ = 0 (RRA = 1), T = 40Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) 100 99.45 99.45 98.84 98.92 98.77 98.89Cost 0% 73.7% 73.6% 222.1% 196.7% 243.3% 206.4%
(c) γ = −5 (RRA = 6), T = 20Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) -100 -134.58 -205.42 -141.00 -194.01 -191.47 -236.86Cost 0% 6.1% 15.5% 7.1% 14.2% 13.9% 18.8%
(d) γ = −5 (RRA = 6), T = 40Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) -100 -202.92 -314.64 -351.06 -545.40 -477.53 -682.83Cost 0% 15.2% 25.8% 28.6% 40.4% 36.7% 46.8%
(e) γ = −11 (RRA = 12), T = 20Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) -100 -192.4 -800.9 -562.1 -3379.7 -1326.5 -5519.4Cost 0% 6.1% 20.8% 17% 37.7% 26.5% 44%
(f) γ = −11 (RRA = 12), T = 40Strategy: Optimal Static Deterministic lifestyle
stochastic S M B-10 B-5 A-10 A-5V (0, 0) -100 -1048 -2066 -25860 -175732 -57602 -309662Cost 0% 23.8% 31.7% 65.7% 97.2% 78.2% 107.7%
Table 1: Relative expected terminal utility for different values of γ and duration ofcontract, T . Expected terminal utilities have been normalised so that the optimalstochastic lifestyle strategy is +100 or -100 (for γ ≥ 0 or γ < 0 respectively). Costis the cost of suboptimality: the relative increase required in the contribution rateto match the optimal expected terminal utility assuming a contribution rate of10% of salary.
33
always gives a better approximation than the determinstic lifestyle strate-gies.) Second, the general increases in the cost of suboptimality for memberswith RRA = 12 reflect the more severe penalties that apply when there iseven a small deviation from the optimal strategy.
• When members have RRA = 1, the static strategies S and M are almostidentical, so that the costs are almost equal.
• When members have RRA = 1, the costs of suboptimality are highly sig-nificant. One reason for this is that the stochastic lifestyling strategy variessubstantially over time, so that even the best, unconstrained static strategylooks very poor. Moreover, the low RRA implies that it is optimal most ofthe time to be very short in cash (even strategies S and M are -100% of thenet pension wealth in cash). Deterministic lifestyle strategies are thereforevery costly because they implicitly have a no-short-selling constraint.26
• When members have RRA = 12, determinstic lifestyle strategies are alsovery poor. The reason is that the initial 100% equity investment is far toohigh for such highly risk-averse individuals.
• When members have the intermediate value of RRA = 6, we see that thecosts of suboptimality are not so great, although they are not negligibleeither. This is not because of any special characteristic of such members. Itis simply that the reasons discussed above explaining why the sub-optimalstrategies are so costly for members with both lower and higher RRA’s donot apply so strongly for members with RRA = 6.
• The salary-hedged static strategy, S, always outperforms significantly thedeterministic lifestyle strategies. This indicates the importance of takinginto account the plan member’s personal salary profile rather than selling astandard ‘one-size-fits-all’ asset allocation strategy.
• The B-5 and B-10 strategies are all superior to the corresponding A-5 andA-10 strategies. This indicates that the plan member’s preference for apension at retirement over a cash lump sum needs to be matched by aswitch into long-dated bonds before retirement (B strategies) rather thancash (A strategies).
• Finally, we note that the determinstic lifestyle strategies are the worst strate-gies of all.
26This constraint can be mitigated somewhat by permitting the use of highly-geared mutualfunds: an investment of 200% of net pension wealth in equities (which is not permissible) (and-100% in cash) might then be equivalent, say, to a 100% investment in a highly-geared fund(which is permissible).
34
4 Conclusions
Stochastic lifestyling has at least two advantages over deterministic lifestyling inrespect of defined contribution pension plans: it takes into account both the planmember’s attitude to risk and the correlation between his salary and asset returns.It is implemented using three efficient mutual funds, resembling investment incash, bonds and equities, respectively. The equity fund is regarded as high riskand its purpose is to satisfy the risk appetite of the plan member. The cash andbond funds are regarded as low risk, but serve different purposes: in particular,the former can be interpreted as a default low-risk investment, whereas the latteris a hedge against annuity-rate risk. In the early stages of the plan, the cashfund is therefore the dominant low-risk component of the investment strategy,but as the retirement date approaches, there is a gradual switch from cash intobonds. This switching between mutual funds superficially resembles deterministiclifestyling. However, instead of switching from high-risk assets to low-risk assets,as in the case of deterministic lifestyling, the optimal stochastic lifestyle strategyinvolves a switch between different types of low-risk assets.
Our results have important practical relevance, since they suggest that the costsof suboptimal policies can be considerable: plan members can (typically) expectto be substantially better off if they adopt a stochastic lifestyling strategy ratherthan a either a static or a deterministic-lifestyling asset-allocation strategy. Ourresults also suggest that many commercial pension plans could be significantly im-proved by making appropriate use of cash and bond investments along stochastic-lifestyling lines. If the investment opportunity set and the plan member’s attitudeto risk are unchanging, all but two of the parameters needed to operate the strat-egy are predetermined and the values of the two stochastic variables required(namely the ratio of current pension wealth to the plan member’s current salaryand the risk-free rate of interest) are easy to measure.
Finally, the following extensions to the current study suggest themselves: (a)developing numerical results for Case 4 with stochastic interest rates, althoughwe conjecture that the results will be similar to those reported in Table 1, (b)allowing for a more general model determining salary growth, e.g., letting µY (t)be stochastic, and (c) considering more general utility functions, such as Epstein& Zin (1989).
Acknowledgements
The authors gratefully acknowledge support from the BSI Gamma Foundation.They also wish to thank participants at the “Risk Theory” and “Stochastic Anal-ysis in Finance and Insurance” meetings in Oberwolfach in September 1999 and
35
May 2000 respectively, for their helpful comments. This includes, in particular,Griselda Deelstra and Ralf Korn. Finally, they wish to thank the two referees ofthe paper for their extremely helpful comments.
References
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37
Appendices
A Proof of Theorem 3.2.1
Let us consider the composition of p∗(t, x, r; V ) in equation (3.2.2) in terms ofthree efficient portfolios or mutual funds which we denote pA, pB, and pC .
Suppose that π = 0. Then dX(t)/X(t) has an expected value of m(p) = p′C(ξ −σY )− µY (t) and an instantaneous variance of v(p) = σ2
Y 0 +σ′Y σY +p′Dp−2p′CσY ,where D = CC ′. Now minimise v(p) over p:
⇒ 2Dp− 2CσY = 0
⇒ p∗(0) = D−1CσY = C ′−1σY ≡ pA, say.
Then m(pA) = σ′Y (ρ− σY )− µY (t) ≡ mA, say.
Next minimise v(p) over p subject to m(p) = m, using the method of Lagrangemultipliers, with L(p, ψ) = v(p) + 2ψ(m(p)−m). Therefore,
∂L
∂p= 2Dp− 2CσY + 2ψC(ξ − σY ) = 0
⇒ p∗(m−mA) = C ′−1[(1 + ψ)σY − ψξ
]∂L
∂ψ= 2(m(p)−m) = 0
⇒ 0 =[CσY − ψC(ξ − σY )
]′D−1C(ξ − σY )− µY (t)−m
⇒ ψ = ψ(m) =mA −m
(ξ − σY )′(ξ − σY ).
The important point to note is that the optimal portfolio p∗(m−mA) is a weightedaverage of the minimum-variance portfolio pA = C ′−1σY (with weight 1 + ψ) andanother efficient portfolio pC = C ′−1ξ (with weight −ψ).
Now consider H(t) = X(t)/a(t, r(t)) where a(t, r(t)) is defined in equation (3.1.1).Since a(t, r(t)) is a function of r(t) only, we can write (using Ito’s formula)
da(t, r(t)) as a(t, r(t))[− da(r)dr(t) + 1
2ca(r)(dr(t))2
], where da(r) is the dura-
tion of the annuity function and ca(r) is its convexity,
da(r) = − 1
a(t, r)
∂a(t, r)
∂r
and ca(r) =1
a(t, r)
∂2a(t, r)
∂r2.
38
Therefore,
dH(t) = H(t)
[(p′C(ξ − σY )− µY (t)) dt− σY 0dZ0(t) + (p′C − σ′Y )dZ(t)
+da(r){µr(r)dt + σr(r)
′dZ(t)}
+
(da(r)
2 − 1
2ca(r)
)σr(r)
′σr(r)dt + da(r) (p′C − σ′Y ) σr(r)dt
]
=D H(t)(m(p, r)dt +
√v(p, r)dZ
),
where Z(t) is a Brownian motion, =D means ‘equivalent in distribution’,
m(p, r) = p′C(ξ − σY )− µY (t) + da(r)µr(r) + da(r)(p′C − σ′Y )σr(r)
+
(da(r)
2 − 1
2ca(r)
)σr(r)
′σr(r)
and v(p, r) = σ2Y 0 + σ′Y σY + p′Dp− 2p′CσY + 2da(r)p
′Cσr(r)
−2da(r)σ′Y σr(r) + da(r)
2σr(r)′σr(r).
Now, minimise v(p, r) over p:27
⇒ 2Dp− 2CσY + 2da(r)Cσr(r) = 0
⇒ p = C ′−1(σY − da(r)σr(r)
)= pB, say.
Let mB = m(pB, r). Next, minimise v(p, r) over p subject to m(p, r) = m. LetpB(m−mB) be the optimal p for this problem. Let L(p, ψ) = v(p, r)+2ψ(m(p, r)−m).
Therefore,
∂L
∂p= 2Dp− 2CσY + 2da(r)Cσr(r) + 2ψC(ξ − σY + da(r)σr(r)) = 0
⇒ p = C ′−1(σY − da(r)σr(r)− ψ
(ξ − σY + da(r)σr(r)
))= (1 + ψ)C ′−1(
σY − da(r)σr(r))− ψC ′−1
ξ
= (1 + ψ)pB − ψpC
∂L
∂ψ= 0
⇒ ψ =mB −m
(ξ − σY + da(r)σr(r))′ (ξ − σY + da(r)σr(r))
.
27Note that pB = pA if da(r) = 0 (that is, if we are funding for a cash lump sum rather thana pension at retirement).
39
As before we see that the optimal asset allocation strategy, pB(mB −m, r), is aweighted average of the minimum variance portfolio, pB, and the more risky, butstill efficient, portfolio, pC , derived earlier.
B Proof of Theorem 3.3.1
That V (t, x, r) is proportional to xγ is clear. For example, suppose we replace x bykx but follow the same asset allocation strategy p(t). We see that J(t, kx, r, p) =kγJ(t, x, r, p) since X(T ) is replaced by kX(T ) if we follow the same strategy.
Now try V = γ−1g1−γxγ in equation (3.3.1). First note that
Vt =(1− γ)
γg−γgtx
γ
Vx = g1−γxγ−1
Vxx = (γ − 1)g1−γxγ−2
Vr =(1− γ)
γg−γgrx
γ
Vrr = (γ − 1)g−γ−1g2rx
γ +(1− γ)
γg−γgrrx
γ
and Vxr = (1− γ)g−γgrxγ−1.
Then (3.2.3) becomes
(1− γ)
γg−γgtx
γ + µr(r)(1− γ)
γg−γgrx
γ
+(− µY (t) + σ′Y (ξ − σY )
)g1−γxγ
+1
2σr(r)
′σr(r)
[(γ − 1)g−γ−1g2
rxγ +
(1− γ)
γg−γgrrx
γ
]
−1
2(ξ − σY )′(ξ − σY )
g2−2γx2γ−2
(γ − 1)g1−γxγ−2
−(ξ − σY )σr(r)g1−γxγ−1(1− γ)g−γgrx
γ−1
(γ − 1)g1−γxγ−2
−1
2σr(r)
′σr(r)(1− γ)2g−2γg2
rx2γ−2
(γ − 1)g1−γxγ−2= 0.
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Now take out g−γ−1xγ as a common factor. Then
(1− γ)
γggt + µr(r)
(1− γ)
γggr + (−µY (t) + σ′Y (ξ − σY ))g2
+1
2σr(r)
′σr(r)
((γ − 1)g2
r +(1− γ)
γggrr
)− 1
2(ξ − σY )′(ξ − σY )
1
(γ − 1)g2
+(ξ − σY )′σr(r)ggr − 1
2(γ − 1)σr(r)
′σr(r)g2r = 0.
The g2r terms cancel, and, with further simplification and division by g, we get
(1− γ)
γgt +
1
2
(1− γ)
γσr(r)
′σr(r)grr +
((1− γ)
γµr(r) + (ξ − σY )′σr(r)
)gr
+
(−µY (t) + σ′Y (ξ − σY )− 1
2(ξ − σY )′(ξ − σY )
1
(γ − 1)
)g = 0.
In the final step we replace µY (t) by µY (t)− σ′Y σY and multiply by γ/(1− γ) toget the PDE for g in equation (3.3.1) in the Theorem.
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